[Math] Prove Simpson’s rule (including error) using the integral remainder

Question

I have to prove Simpson's rule including the error with the help of the integral remainder. However, I have practically no idea how to start.

Let $f: [a,b] \rightarrow \mathbb{R}$ be continuously differentiable four times. Let $n \in 2\mathbb{N}$ and $h = (b-a)/n$. Let $y_k = f(a+kh)$ for $0 \leq k \leq n$. Show:

$$\left|\int_a^b f(x) \mathrm dx – \frac{h}{3} [(y_0 + y_n) + 4(y_1+y_3+…+y_{n-1})+2(y_2+y_4+…+y_{n-2})]\right|$$
$$\leq \frac{1}{180} \max_{a \leq x \leq b} |f^{(4)}(x)|(b-a)h^4$$

I thought I'd do a Taylor expansion for $\int_a^b f(x) \mathrm dx$ and then apply the integral remainder, but then all terms vanish and I get $0=0$ which is not very helpful…

I only need a good advice how to start! Thanks in advance!

Answer 1

A nice, very readable reference for this question is a recent paper in the Monthly, "Simpson's rule is exact for quintics". It is available for download at the author's (Louis A. Talman) webpage. It appeared American Mathematical Monthly, 113(2006), 144–155. I recommend it; I have used it a couple of times when covering this material in class. Here is the abstract:

In this article, we use tools accessible to freshman calculus students to develop exact—though usually uncomputable—expressions for the error that results in replacing a definite integral with its midpoint rule, trapezoidal rule, or Simpson's rule approximation. Among the tools we use is an extended version of the first mean value theorem for integrals. We obtain not only the classical estimates that appear in calculus books, but estimates for functions less smooth than the classical results require. We show, in particular, how to compute the exact error for a Simpson's rule approximation to an integral of a quintic polynomial.

If you visit Talman's page, you may also enjoy taking a look at the nice "The Mother of All Calculus Quizzes."